1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,280 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,280 --> 00:00:18,450 at ocw.mit.edu. 8 00:00:26,127 --> 00:00:28,710 DENNIS FREEMAN: Everybody would like to believe that they just 9 00:00:28,710 --> 00:00:31,279 simply sort of think about the subject a little bit, 10 00:00:31,279 --> 00:00:32,320 and it will come to them. 11 00:00:32,320 --> 00:00:35,160 It's kind of like the osmosis theory of learning. 12 00:00:35,160 --> 00:00:37,350 But in fact, the way you get good at something 13 00:00:37,350 --> 00:00:38,550 is practicing. 14 00:00:38,550 --> 00:00:39,840 It's the same for music. 15 00:00:39,840 --> 00:00:40,980 It's the same for sports. 16 00:00:40,980 --> 00:00:43,315 It's the same for academics. 17 00:00:43,315 --> 00:00:44,940 So the way you get good in this subject 18 00:00:44,940 --> 00:00:46,640 is to work on the homework. 19 00:00:46,640 --> 00:00:50,650 We'll give two kinds of homework. 20 00:00:50,650 --> 00:00:53,320 There will be conventional kinds of problems followed 21 00:00:53,320 --> 00:00:56,680 by engineering design problems. 22 00:00:56,680 --> 00:00:58,110 The conventional kinds of problems 23 00:00:58,110 --> 00:01:01,350 are intended to be the kinds of problems 24 00:01:01,350 --> 00:01:08,390 that have simple, unambiguous, easily checkable answers-- 25 00:01:08,390 --> 00:01:15,860 so well-defined problem, single, unambiguous answer, 26 00:01:15,860 --> 00:01:17,930 the kinds of problems that you might be expecting 27 00:01:17,930 --> 00:01:22,080 to see on an exam, for example. 28 00:01:22,080 --> 00:01:25,874 But then we'll also have engineering design problems. 29 00:01:25,874 --> 00:01:27,540 Engineering design problems are intended 30 00:01:27,540 --> 00:01:31,710 to be a little more ambitious, a little more 31 00:01:31,710 --> 00:01:38,690 fun, a little harder, perhaps, perhaps not completely 32 00:01:38,690 --> 00:01:41,310 well-specified, sort of more like the kind of a problem 33 00:01:41,310 --> 00:01:44,700 that your boss might give you after you've graduated. 34 00:01:44,700 --> 00:01:47,670 So part of the exercise will be figuring out 35 00:01:47,670 --> 00:01:51,240 exactly how to convince somebody that you've 36 00:01:51,240 --> 00:01:53,260 got the right answer. 37 00:01:53,260 --> 00:01:56,880 So those kinds of problems are often-- 38 00:01:56,880 --> 00:01:59,970 it will often be the case that numerics will help you 39 00:01:59,970 --> 00:02:02,600 in making an argument. 40 00:02:02,600 --> 00:02:04,630 So it often will be the case that we'll 41 00:02:04,630 --> 00:02:06,370 ask you to plot something. 42 00:02:06,370 --> 00:02:09,880 Or maybe you just decide that a plot is the most effective way 43 00:02:09,880 --> 00:02:12,140 to communicate your result. 44 00:02:12,140 --> 00:02:15,830 We are completely ambivalent about what programming 45 00:02:15,830 --> 00:02:17,750 language that you use. 46 00:02:17,750 --> 00:02:21,170 Since 6.01 and 6.02 are prerequisites, 47 00:02:21,170 --> 00:02:23,120 we assume you all know Python. 48 00:02:23,120 --> 00:02:26,780 So all the examples that I give in lecture handouts 49 00:02:26,780 --> 00:02:30,080 or in homework solutions will be in Python. 50 00:02:30,080 --> 00:02:33,170 But if you'd rather use Matlab, I just don't care. 51 00:02:33,170 --> 00:02:36,260 OK, we're completely ambivalent about what programming language 52 00:02:36,260 --> 00:02:36,940 you use. 53 00:02:36,940 --> 00:02:39,560 But it will be important that in some of the problems 54 00:02:39,560 --> 00:02:44,540 you'll find that useful to be able to generate 55 00:02:44,540 --> 00:02:48,200 a numerical kind of solution. 56 00:02:48,200 --> 00:02:50,074 These are very different kinds of problems. 57 00:02:50,074 --> 00:02:51,490 And to help you with both of them, 58 00:02:51,490 --> 00:02:55,220 we have a different idea about each. 59 00:02:55,220 --> 00:02:58,170 So to help you with the conventional problems, the kind 60 00:02:58,170 --> 00:03:02,520 that have precise, easy-to-check answers, 61 00:03:02,520 --> 00:03:05,340 we've developed for the first time ever over the summer 62 00:03:05,340 --> 00:03:09,450 a tutor-type environment for 6.003. 63 00:03:09,450 --> 00:03:11,850 So this was intended to be like the tutor environment 64 00:03:11,850 --> 00:03:13,500 that we use in 6.01. 65 00:03:13,500 --> 00:03:16,147 It's one of the more popular aspects of 6.01, 66 00:03:16,147 --> 00:03:17,730 the fact that you can put in an answer 67 00:03:17,730 --> 00:03:20,460 and hit the button that says Check. 68 00:03:20,460 --> 00:03:23,160 OK, so we're going to have one of those. 69 00:03:23,160 --> 00:03:26,510 So the idea is going to be that in the case that we ask you 70 00:03:26,510 --> 00:03:31,380 a simple question that has a precise answer, 71 00:03:31,380 --> 00:03:34,290 we'll use a tutor environment to let you check your answer 72 00:03:34,290 --> 00:03:36,480 to see if you got the right answer. 73 00:03:36,480 --> 00:03:38,610 In the case of the engineering design problems, 74 00:03:38,610 --> 00:03:39,570 that won't be the case. 75 00:03:39,570 --> 00:03:42,360 It will not be the case that it's very easy at all 76 00:03:42,360 --> 00:03:45,974 to think about a computer program that would check you. 77 00:03:45,974 --> 00:03:47,890 So the alternative is going to be that we will 78 00:03:47,890 --> 00:03:50,847 have extended office hours. 79 00:03:50,847 --> 00:03:53,180 So we're going to have something that I think of as open 80 00:03:53,180 --> 00:03:55,390 or block office hours. 81 00:03:55,390 --> 00:03:59,380 Monday and Tuesday afternoon and early evening, 82 00:03:59,380 --> 00:04:03,010 we've reserved the basement of Stata, 32-044, 83 00:04:03,010 --> 00:04:06,250 for use of this class, the idea being that that's 84 00:04:06,250 --> 00:04:07,750 a nice space you can just come there 85 00:04:07,750 --> 00:04:10,570 to work any time you want to 86 00:04:10,570 --> 00:04:14,720 hopefully because it's nice, and hopefully because you all 87 00:04:14,720 --> 00:04:18,310 know that there will be other 6.003 people there, 88 00:04:18,310 --> 00:04:21,070 you'll just decide that's a good place to work. 89 00:04:21,070 --> 00:04:23,680 And that's convenient because in these engineering design 90 00:04:23,680 --> 00:04:26,050 problems, if you run into something that's 91 00:04:26,050 --> 00:04:27,970 unclear to you, we don't really want 92 00:04:27,970 --> 00:04:30,660 you to spend an hour pondering what was being asked. 93 00:04:33,370 --> 00:04:36,950 So we want to make it easy for you to get help. 94 00:04:36,950 --> 00:04:40,330 And you can get help from peers, from other students who 95 00:04:40,330 --> 00:04:43,150 are there, but we'll also make sure 96 00:04:43,150 --> 00:04:45,730 that there is at least one 6.003 staff member there all 97 00:04:45,730 --> 00:04:47,306 the time. 98 00:04:47,306 --> 00:04:50,390 OK, so the homeworks will be due on Wednesday. 99 00:04:50,390 --> 00:04:53,930 That's the strategic value of Mondays and Tuesdays. 100 00:04:53,930 --> 00:04:58,150 So you can use that time to get ready for turning 101 00:04:58,150 --> 00:05:00,604 in the homework on Wednesday. 102 00:05:00,604 --> 00:05:01,520 Questions or comments? 103 00:05:05,830 --> 00:05:08,280 OK, we're required to say something 104 00:05:08,280 --> 00:05:11,640 about what we mean by a collaboration policy. 105 00:05:11,640 --> 00:05:14,522 So in this course, we would like you to talk to each other 106 00:05:14,522 --> 00:05:16,480 and help each other figure out what's going on. 107 00:05:19,390 --> 00:05:21,730 At the same time, we'd like to reward you 108 00:05:21,730 --> 00:05:26,860 for being on the honest side, by which I'm going to mean, 109 00:05:26,860 --> 00:05:29,250 if you sign your name to something, you actually did it. 110 00:05:32,160 --> 00:05:34,950 So we'd like to believe that the homework that you turn in 111 00:05:34,950 --> 00:05:38,040 under your name, you did. 112 00:05:38,040 --> 00:05:40,570 You're perfectly welcome to talk to each other, 113 00:05:40,570 --> 00:05:42,900 to talk to the staff, to talk to friends who 114 00:05:42,900 --> 00:05:44,490 took this course previously. 115 00:05:44,490 --> 00:05:45,990 Get all the help you like in trying 116 00:05:45,990 --> 00:05:47,327 to understand the concepts. 117 00:05:47,327 --> 00:05:49,410 But when it comes down to writing up your homework 118 00:05:49,410 --> 00:05:52,860 and turning it in, we expect that what you signed your name 119 00:05:52,860 --> 00:05:55,590 to is something you did. 120 00:05:55,590 --> 00:05:58,410 If you got especially large amount of help 121 00:05:58,410 --> 00:06:02,490 at the conceptual level, we'd like you to acknowledge that. 122 00:06:02,490 --> 00:06:05,250 You don't need to acknowledge that the TAs helped you. 123 00:06:05,250 --> 00:06:06,919 We expect that. 124 00:06:06,919 --> 00:06:08,460 But if you collaborated with somebody 125 00:06:08,460 --> 00:06:10,530 to understand what the problem was, 126 00:06:10,530 --> 00:06:13,050 we'd like you to tell us that. 127 00:06:13,050 --> 00:06:15,390 Just write at the top, collaborated on figuring out 128 00:06:15,390 --> 00:06:18,260 the concepts with so-and-so. 129 00:06:18,260 --> 00:06:21,110 But we'd like you to have written your homework. 130 00:06:21,110 --> 00:06:23,810 So for example, we would like you not to say, 131 00:06:23,810 --> 00:06:26,780 I copied so-and-so's is homework word for word 132 00:06:26,780 --> 00:06:29,510 even though that's an honest statement, that's not something 133 00:06:29,510 --> 00:06:31,580 that we're actually looking for. 134 00:06:31,580 --> 00:06:32,841 OK, is that clear? 135 00:06:32,841 --> 00:06:34,340 We want you to work together, but we 136 00:06:34,340 --> 00:06:38,470 would like you to write up your own solutions. 137 00:06:38,470 --> 00:06:44,130 OK, we have firm deadlines. 138 00:06:44,130 --> 00:06:48,300 The homeworks will always be due on Wednesday. 139 00:06:48,300 --> 00:06:52,010 You'll be able to slip on one without penalty. 140 00:06:52,010 --> 00:06:54,770 So if you turn in one late, it won't change-- 141 00:06:54,770 --> 00:06:57,360 it won't have any effect on your grade whatever. 142 00:06:57,360 --> 00:07:00,910 If you do twice, that could have an effect. 143 00:07:00,910 --> 00:07:04,060 Unless you're excused by a dean or an instructor 144 00:07:04,060 --> 00:07:08,295 or a medical person, that submission 145 00:07:08,295 --> 00:07:10,170 will count half what it would normally count. 146 00:07:12,654 --> 00:07:15,070 I should have mentioned when I was talking about homework, 147 00:07:15,070 --> 00:07:16,690 homework 1 is already posted. 148 00:07:16,690 --> 00:07:18,604 It will be due next Wednesday. 149 00:07:21,330 --> 00:07:27,060 The online submission part is almost ready to be posted 150 00:07:27,060 --> 00:07:29,110 and will be posted later this afternoon. 151 00:07:29,110 --> 00:07:32,070 So right now, you can see all the problems. 152 00:07:32,070 --> 00:07:34,997 But the online will become available this afternoon. 153 00:07:38,820 --> 00:07:42,840 OK, here is 003 at a glance. 154 00:07:42,840 --> 00:07:46,500 It's not too unlike what you might have expected. 155 00:07:46,500 --> 00:07:50,280 Just a couple of things to point out, the order of coverage 156 00:07:50,280 --> 00:07:52,900 is not the same as the order of coverage 157 00:07:52,900 --> 00:07:56,370 in Oppenheim and Willsky. 158 00:07:56,370 --> 00:07:58,350 The reason for doing that is that having taught 159 00:07:58,350 --> 00:08:00,090 this class for 15 years, I just think 160 00:08:00,090 --> 00:08:05,120 there's some subjects that are easier starting points 161 00:08:05,120 --> 00:08:07,150 than others. 162 00:08:07,150 --> 00:08:09,160 So I'm doing what I think is the easiest 163 00:08:09,160 --> 00:08:12,340 entry to this material with the idea 164 00:08:12,340 --> 00:08:14,620 that after you've got the easy stuff under your belt, 165 00:08:14,620 --> 00:08:17,860 it's easier to move on to the more difficult material. 166 00:08:17,860 --> 00:08:19,630 So the order is not precisely the same 167 00:08:19,630 --> 00:08:21,940 as that in Oppenheim and Willsky. 168 00:08:21,940 --> 00:08:25,360 But there is a map on the website that gives you 169 00:08:25,360 --> 00:08:29,320 a week-by-week breakdown of what part of Oppenheim and Willsky 170 00:08:29,320 --> 00:08:32,120 are we working on right now. 171 00:08:32,120 --> 00:08:35,870 So Oppenheim and Willsky is the recommended text. 172 00:08:35,870 --> 00:08:37,600 But it's not quite in the regular order 173 00:08:37,600 --> 00:08:38,641 of Oppenheim and Willsky. 174 00:08:38,641 --> 00:08:42,250 See We will eventually cover exactly the same material that 175 00:08:42,250 --> 00:08:43,669 is in-- 176 00:08:43,669 --> 00:08:46,130 Oppenheim and Willsky was written for this class. 177 00:08:46,130 --> 00:08:48,220 We'll cover the same material, just 178 00:08:48,220 --> 00:08:49,720 not quite in the same order. 179 00:08:52,870 --> 00:08:54,630 Another thing to notice is exams. 180 00:08:54,630 --> 00:08:58,960 We'll have three midterm exams and a final. 181 00:08:58,960 --> 00:09:02,035 The midterm exams are all evening exams, 7:30 to 9:30 182 00:09:02,035 --> 00:09:02,910 on Wednesday evening. 183 00:09:05,940 --> 00:09:09,300 The idea is that there's intended 184 00:09:09,300 --> 00:09:11,700 to be a gentle ramp through the exams. 185 00:09:11,700 --> 00:09:13,200 Exam 1 is worth 10%. 186 00:09:13,200 --> 00:09:15,240 Exam 2 is worth 15%. 187 00:09:15,240 --> 00:09:17,400 Exam 3 is worth 20%. 188 00:09:17,400 --> 00:09:20,390 The final exam is worth 40% 189 00:09:20,390 --> 00:09:24,789 so the idea is that if you have a mismatch on the way-- 190 00:09:24,789 --> 00:09:26,330 if you don't quite understand the way 191 00:09:26,330 --> 00:09:27,440 we're going to be asking questions 192 00:09:27,440 --> 00:09:29,210 and so forth, there's not a big penalty 193 00:09:29,210 --> 00:09:31,160 for screwing up the first exam. 194 00:09:31,160 --> 00:09:31,890 That's the idea. 195 00:09:31,890 --> 00:09:34,940 So there's intended to be a gentle ramp 196 00:09:34,940 --> 00:09:37,672 in how much of the exams count. 197 00:09:41,520 --> 00:09:44,510 OK, and finally, I'd like feedback 198 00:09:44,510 --> 00:09:48,350 on what you think is happening. 199 00:09:48,350 --> 00:09:50,960 So I would like to ask for a few volunteers-- 200 00:09:50,960 --> 00:09:52,640 few is four or fewer-- 201 00:09:55,500 --> 00:09:57,420 who would be interested to meet with me 202 00:09:57,420 --> 00:10:00,730 once a week just to tell me what you think. 203 00:10:00,730 --> 00:10:04,030 Ideally, those people would be kind of outgoing 204 00:10:04,030 --> 00:10:07,241 and tell me things that other people think too. 205 00:10:07,241 --> 00:10:08,740 But if you're completely introverted 206 00:10:08,740 --> 00:10:12,520 and want to only tell me what you think, that's fine too. 207 00:10:12,520 --> 00:10:16,540 So this is an opportunity for you 208 00:10:16,540 --> 00:10:20,241 to tell the staff, to give us feedback 209 00:10:20,241 --> 00:10:21,240 on how things are going. 210 00:10:21,240 --> 00:10:22,115 We're going too slow. 211 00:10:22,115 --> 00:10:23,050 We're going too fast. 212 00:10:23,050 --> 00:10:24,008 We're going too boring. 213 00:10:24,008 --> 00:10:26,565 We're intensely too interesting, that kind of stuff. 214 00:10:29,710 --> 00:10:35,350 And it's an opportunity for you to understand our perspective. 215 00:10:35,350 --> 00:10:37,370 Well, the reason we did that was blah. 216 00:10:37,370 --> 00:10:40,180 And you can tell us, yeah, but that's not important. 217 00:10:40,180 --> 00:10:46,170 So it's an opportunity for you to convey to us 218 00:10:46,170 --> 00:10:48,420 how you think things are going and how you think 219 00:10:48,420 --> 00:10:49,740 things should be different. 220 00:10:49,740 --> 00:10:52,410 It's also an opportunity for you to learn about teaching. 221 00:10:52,410 --> 00:10:55,020 So if you think about teaching as a career, 222 00:10:55,020 --> 00:10:56,970 it's a good thing to do-- 223 00:10:56,970 --> 00:10:59,730 completely voluntary, has absolutely no effect 224 00:10:59,730 --> 00:11:02,110 on your grade. 225 00:11:02,110 --> 00:11:04,420 If you think you might be interested, 226 00:11:04,420 --> 00:11:06,271 it's probably going to meet on-- 227 00:11:06,271 --> 00:11:08,020 we'll probably meet on Thursday afternoon. 228 00:11:08,020 --> 00:11:11,085 But that's negotiable since it's only four people. 229 00:11:11,085 --> 00:11:13,210 And if you are interested, please send me an email. 230 00:11:13,210 --> 00:11:16,100 And the first four people who ask, they're there. 231 00:11:19,168 --> 00:11:21,976 OK? 232 00:11:21,976 --> 00:11:25,892 That's it on administration. 233 00:11:25,892 --> 00:11:27,350 Are there things you'd like to know 234 00:11:27,350 --> 00:11:28,891 about course administration before we 235 00:11:28,891 --> 00:11:30,200 go on to technical things? 236 00:11:35,430 --> 00:11:36,460 Wonderful. 237 00:11:36,460 --> 00:11:39,570 OK, this course is about systems. 238 00:11:42,450 --> 00:11:45,990 This is not the first course you've had about systems. 239 00:11:45,990 --> 00:11:47,697 You've had lots of courses in systems. 240 00:11:47,697 --> 00:11:50,030 Every physics course you've ever taken is about systems. 241 00:11:53,530 --> 00:11:55,030 What's different about this course 242 00:11:55,030 --> 00:11:57,280 is the way we think about systems. 243 00:11:57,280 --> 00:11:59,500 We think about systems in a particular way that 244 00:11:59,500 --> 00:12:03,610 turns out to be extraordinarily powerful, useful, 245 00:12:03,610 --> 00:12:05,850 shows up all the time. 246 00:12:05,850 --> 00:12:07,980 And that's what the focus of the course is. 247 00:12:07,980 --> 00:12:08,480 That 248 00:12:08,480 --> 00:12:11,130 The abstraction that we're going to use 249 00:12:11,130 --> 00:12:16,620 is we think about a system as a thing that 250 00:12:16,620 --> 00:12:19,610 has an input and an output. 251 00:12:19,610 --> 00:12:24,100 Given the input and the system, you can compute the output. 252 00:12:24,100 --> 00:12:27,430 Systems have one input and one output. 253 00:12:27,430 --> 00:12:30,660 It's a very special way of thinking 254 00:12:30,660 --> 00:12:36,670 about a system that turns out to be surprisingly powerful. 255 00:12:36,670 --> 00:12:39,000 It's so powerful that that's really the thing that we 256 00:12:39,000 --> 00:12:40,280 will focus on in this class. 257 00:12:40,280 --> 00:12:53,860 So it's the thing that I will call the 6.003 abstraction. 258 00:12:53,860 --> 00:12:57,610 That's going to be the theme in everything we do, 259 00:12:57,610 --> 00:13:00,190 that representation. 260 00:13:00,190 --> 00:13:02,020 And the best way to get across, I think, 261 00:13:02,020 --> 00:13:04,180 how that's different from other representations 262 00:13:04,180 --> 00:13:05,890 that you're already quite familiar with 263 00:13:05,890 --> 00:13:08,330 is to just look at an example. 264 00:13:08,330 --> 00:13:12,040 So here's a system that I'm sure you all know all about. 265 00:13:12,040 --> 00:13:12,705 Right? 266 00:13:12,705 --> 00:13:14,080 This is not the first time you've 267 00:13:14,080 --> 00:13:15,204 seen this kind of a system. 268 00:13:17,740 --> 00:13:20,530 So you all know how to do this. 269 00:13:20,530 --> 00:13:22,945 Free-body diagrams, F equals ma-- 270 00:13:22,945 --> 00:13:25,120 there's an enormous number of ways 271 00:13:25,120 --> 00:13:26,700 that you can analyze such a system. 272 00:13:26,700 --> 00:13:30,550 We're going to take a very special approach 273 00:13:30,550 --> 00:13:34,340 where we think about the system as having 274 00:13:34,340 --> 00:13:36,400 an input and an output. 275 00:13:39,090 --> 00:13:41,960 OK, that's a little bit arbitrary. 276 00:13:41,960 --> 00:13:44,400 We'll say a little more in a minute about how arbitrary 277 00:13:44,400 --> 00:13:46,180 it is. 278 00:13:46,180 --> 00:13:49,200 But the idea is going to be that this system, 279 00:13:49,200 --> 00:13:52,260 the mass-spring system, the anvil with a spring, 280 00:13:52,260 --> 00:13:56,060 there's an input, and there's an output. 281 00:13:56,060 --> 00:13:58,810 So for example, for some particular purpose, 282 00:13:58,810 --> 00:14:01,730 I might want to think that the input is 283 00:14:01,730 --> 00:14:05,070 the position of my hand. 284 00:14:05,070 --> 00:14:07,160 One reason I might want to think about that is it 285 00:14:07,160 --> 00:14:09,120 is under my control. 286 00:14:09,120 --> 00:14:10,890 So it might be reasonable to think 287 00:14:10,890 --> 00:14:13,140 about my hand being the input. 288 00:14:13,140 --> 00:14:16,350 So for example, my hand can do this. 289 00:14:16,350 --> 00:14:17,390 Right? 290 00:14:17,390 --> 00:14:20,600 So I might think about the position of my hand 291 00:14:20,600 --> 00:14:22,670 being the input. 292 00:14:22,670 --> 00:14:26,410 So if I did that, then what I could do 293 00:14:26,410 --> 00:14:30,325 is characterize the input by some waveform, a signal. 294 00:14:35,290 --> 00:14:37,330 In order to use the abstraction, I also 295 00:14:37,330 --> 00:14:39,550 have to say what the output is. 296 00:14:39,550 --> 00:14:41,630 I might be interested, for example, 297 00:14:41,630 --> 00:14:44,310 in knowing the position of the mass. 298 00:14:44,310 --> 00:14:48,740 Then the output would be the displacement of the mass. 299 00:14:48,740 --> 00:14:51,990 That assignment of input and output is somewhat arbitrary. 300 00:14:54,600 --> 00:14:58,420 I get to choose that to make the problem that I'm 301 00:14:58,420 --> 00:15:02,300 interested in as easy as possible. 302 00:15:02,300 --> 00:15:05,810 So for example, if I'm thinking about this mass and spring 303 00:15:05,810 --> 00:15:09,770 system here, using my hand as the input, 304 00:15:09,770 --> 00:15:11,690 that seems kind of natural because that's 305 00:15:11,690 --> 00:15:12,650 the thing I control. 306 00:15:12,650 --> 00:15:14,900 Thinking about the position of the mass as the output, 307 00:15:14,900 --> 00:15:17,690 that seems reasonable because that's 308 00:15:17,690 --> 00:15:19,800 the thing you can take a picture of. 309 00:15:19,800 --> 00:15:23,110 It's the thing you can observe from where you are now. 310 00:15:23,110 --> 00:15:24,640 But it's not unique. 311 00:15:24,640 --> 00:15:27,580 I might have wanted to think about the force 312 00:15:27,580 --> 00:15:32,010 that I'm putting into it as in the input. 313 00:15:32,010 --> 00:15:34,830 I might have wanted to think about the output being 314 00:15:34,830 --> 00:15:36,950 the acceleration of the mass. 315 00:15:36,950 --> 00:15:39,780 Maybe there's some limit to how much acceleration 316 00:15:39,780 --> 00:15:43,190 some mass can take. 317 00:15:43,190 --> 00:15:47,270 So you get to choose the input and the output. 318 00:15:47,270 --> 00:15:50,120 But you have to choose an input and an output. 319 00:15:50,120 --> 00:15:52,190 That's part of the abstraction. 320 00:15:52,190 --> 00:15:53,420 And that's part of the power. 321 00:15:53,420 --> 00:15:56,570 We'll see later that part of the power of doing it this way 322 00:15:56,570 --> 00:16:01,780 is that you draw attention to the things of interest 323 00:16:01,780 --> 00:16:05,090 and push into the background the parts that are not of interest, 324 00:16:05,090 --> 00:16:05,920 at least not now. 325 00:16:08,700 --> 00:16:12,910 This kind of an abstraction is an extremely versatile. 326 00:16:12,910 --> 00:16:17,000 Here's a completely different kind of problem. 327 00:16:17,000 --> 00:16:19,120 Imagine water flowing into a tank. 328 00:16:19,120 --> 00:16:20,980 The tank is leaky, so it flows out. 329 00:16:23,540 --> 00:16:25,027 There's a second tank. 330 00:16:25,027 --> 00:16:26,360 OK, what on earth could this be? 331 00:16:26,360 --> 00:16:29,500 This could be-- maybe this is the-- 332 00:16:29,500 --> 00:16:30,660 it's appropriate today. 333 00:16:30,660 --> 00:16:32,870 It's raining, or it was raining. 334 00:16:35,480 --> 00:16:36,500 It was pouring. 335 00:16:36,500 --> 00:16:39,720 Rain When I came in, it was pouring. 336 00:16:39,720 --> 00:16:46,770 Maybe this is nature delivering rain. 337 00:16:46,770 --> 00:16:50,390 This is the Woburn reservoir from which 338 00:16:50,390 --> 00:16:52,929 we get water in Boston. 339 00:16:52,929 --> 00:16:54,970 Maybe this is the Fresh Pond reservoir from which 340 00:16:54,970 --> 00:16:57,630 we get water in Cambridge. 341 00:16:57,630 --> 00:17:00,900 Maybe this is the rate at which water 342 00:17:00,900 --> 00:17:05,940 goes from Woburn to Fresh Pond. 343 00:17:05,940 --> 00:17:08,730 Maybe this is the rate at which water leaves Fresh Pond 344 00:17:08,730 --> 00:17:12,490 and goes into consumers' houses. 345 00:17:12,490 --> 00:17:15,460 The point is that there's some abstraction. 346 00:17:15,460 --> 00:17:17,260 There's some physical-- actually, 347 00:17:17,260 --> 00:17:18,460 I didn't say that right. 348 00:17:18,460 --> 00:17:21,200 There's a physical thing-- 349 00:17:21,200 --> 00:17:26,660 rain, Woburn, Woburn reservoir, Fresh Pond reservoir, 350 00:17:26,660 --> 00:17:28,314 consumers' houses. 351 00:17:28,314 --> 00:17:29,480 There's some physical thing. 352 00:17:29,480 --> 00:17:32,180 There's some laws of physics that dictate things. 353 00:17:32,180 --> 00:17:37,040 But we ignore-- or ignore is not quite the right way 354 00:17:37,040 --> 00:17:38,150 to think about it. 355 00:17:38,150 --> 00:17:40,850 We think about those rules of physics 356 00:17:40,850 --> 00:17:44,840 as rules that govern the input-output relationship. 357 00:17:44,840 --> 00:17:48,440 So we think about then the entire system. 358 00:17:48,440 --> 00:17:51,950 Rather than thinking about it as reservoirs and rain and water 359 00:17:51,950 --> 00:17:53,630 and flow and all that sort of thing, 360 00:17:53,630 --> 00:17:59,780 we think about it as there's a signal in, rain, 361 00:17:59,780 --> 00:18:03,830 and there's a signal out, water usage. 362 00:18:03,830 --> 00:18:06,290 And we take all the details of the system 363 00:18:06,290 --> 00:18:08,400 and bury it in this box. 364 00:18:08,400 --> 00:18:09,480 It's an abstraction. 365 00:18:09,480 --> 00:18:14,150 It's a way to suppress some details to highlight others. 366 00:18:17,420 --> 00:18:21,710 OK, and we use this in a huge variety of situations. 367 00:18:21,710 --> 00:18:24,820 We can think about a third example here, our cell phone. 368 00:18:24,820 --> 00:18:28,070 Cell phone is an enormously complicated system. 369 00:18:28,070 --> 00:18:30,470 But for the purpose of understanding the input-output 370 00:18:30,470 --> 00:18:35,720 characteristics, if what I really want to know is how good 371 00:18:35,720 --> 00:18:37,700 is the quality of the audio-- 372 00:18:37,700 --> 00:18:40,250 not that that's even a little bit important for cell phones. 373 00:18:40,250 --> 00:18:41,000 Right? 374 00:18:41,000 --> 00:18:43,754 Cell phones are important for texting and taking pictures 375 00:18:43,754 --> 00:18:45,170 and have nothing to do with voice. 376 00:18:45,170 --> 00:18:47,960 But in the old days, cell phones had to do with voice. 377 00:18:47,960 --> 00:18:50,150 And I made this slide back when that was still true. 378 00:18:50,150 --> 00:18:53,560 So I apologize it's out of date. 379 00:18:53,560 --> 00:18:55,970 So here, the important thing of the cell phone system 380 00:18:55,970 --> 00:18:59,540 was the sound in and the sound out. 381 00:18:59,540 --> 00:19:02,750 And the idea was to represent the cell phone system 382 00:19:02,750 --> 00:19:07,300 by the transformation between how the sound comes in 383 00:19:07,300 --> 00:19:10,140 and how the sound comes out. 384 00:19:10,140 --> 00:19:13,170 OK, so we do this for an enormous number of reasons. 385 00:19:13,170 --> 00:19:15,870 One is that it's widely applicable. 386 00:19:15,870 --> 00:19:18,680 You can do this kind of a characterization for systems 387 00:19:18,680 --> 00:19:21,680 from electrical systems, mechanical systems, 388 00:19:21,680 --> 00:19:24,000 optical systems, acoustic systems, biological systems, 389 00:19:24,000 --> 00:19:29,250 financial systems, all over the place. 390 00:19:29,250 --> 00:19:31,380 That means that this kind of an approach 391 00:19:31,380 --> 00:19:33,920 is very powerful because it's so widely applicable. 392 00:19:33,920 --> 00:19:37,860 It also means that, for example, every engineering discipline 393 00:19:37,860 --> 00:19:41,910 everywhere has some course like this. 394 00:19:41,910 --> 00:19:45,690 It's just too powerful not to do it. 395 00:19:45,690 --> 00:19:49,190 So regardless of what department you were in, 396 00:19:49,190 --> 00:19:51,090 in the School of Engineering, for example, 397 00:19:51,090 --> 00:19:52,820 there is something like this. 398 00:19:56,740 --> 00:19:59,470 Another reason this is interesting 399 00:19:59,470 --> 00:20:03,070 is that it provides a layer of abstraction that lets 400 00:20:03,070 --> 00:20:04,760 you focus on certain things. 401 00:20:04,760 --> 00:20:08,280 So for example, let's expand on the cell phone network. 402 00:20:08,280 --> 00:20:10,670 Imagine that sound comes into a cell phone. 403 00:20:10,670 --> 00:20:15,990 Well, the job of the phone is to communicate to a tower. 404 00:20:15,990 --> 00:20:21,090 And that communication is via some electromagnetic signal. 405 00:20:21,090 --> 00:20:25,200 Then the job of the tower is to communicate with another tower. 406 00:20:25,200 --> 00:20:27,629 That happens in all kinds of ways. 407 00:20:27,629 --> 00:20:28,920 It could be a fiber optic link. 408 00:20:28,920 --> 00:20:29,940 It could be a satellite link. 409 00:20:29,940 --> 00:20:31,170 It could be electromagnetism. 410 00:20:31,170 --> 00:20:33,040 It could be lots of different things. 411 00:20:33,040 --> 00:20:35,550 There's a lot of different technologies for getting towers 412 00:20:35,550 --> 00:20:37,290 to talk to towers. 413 00:20:37,290 --> 00:20:39,630 Then towers talk back to cell phones 414 00:20:39,630 --> 00:20:42,780 via electromagnetic signals just like these. 415 00:20:42,780 --> 00:20:47,340 And the cell phone eventually generates an acoustical output. 416 00:20:47,340 --> 00:20:51,000 The idea is that by thinking about the signals and systems 417 00:20:51,000 --> 00:20:56,580 approach, we have abstracted away everything other 418 00:20:56,580 --> 00:20:58,800 than the flow of information. 419 00:20:58,800 --> 00:21:02,780 It makes it easy to concentrate, to follow 420 00:21:02,780 --> 00:21:07,170 the flow of information through a complex system. 421 00:21:07,170 --> 00:21:12,250 Because we have pushed every element 422 00:21:12,250 --> 00:21:14,150 into the same framework-- 423 00:21:14,150 --> 00:21:16,510 every element has an input and an output-- 424 00:21:16,510 --> 00:21:21,610 regardless of the underlying substrate, 425 00:21:21,610 --> 00:21:24,400 the analysis is similar. 426 00:21:24,400 --> 00:21:27,250 That means that components that are 427 00:21:27,250 --> 00:21:29,320 characterized using this abstraction 428 00:21:29,320 --> 00:21:32,320 are easily combined. 429 00:21:32,320 --> 00:21:34,060 So we refer to this as-- 430 00:21:34,060 --> 00:21:37,330 we say that these systems are combinational. 431 00:21:37,330 --> 00:21:40,000 Their combinational in the same sense that Python 432 00:21:40,000 --> 00:21:42,860 was combinational in 6.01. 433 00:21:42,860 --> 00:21:47,500 If you represent things by Python functions, 434 00:21:47,500 --> 00:21:52,850 it's very easy to combine the functions to have a bigger 435 00:21:52,850 --> 00:21:57,150 function whose purpose and details can be understood 436 00:21:57,150 --> 00:22:02,610 without knowing what was inside every individual function. 437 00:22:02,610 --> 00:22:05,730 Similarly here, by knowing the input behavior-- 438 00:22:05,730 --> 00:22:07,290 the input-output behavior of the cell 439 00:22:07,290 --> 00:22:10,590 phone, the input-output behavior of the tower, 440 00:22:10,590 --> 00:22:12,270 the input-output behavior of this tower, 441 00:22:12,270 --> 00:22:13,650 it's easy to compose then. 442 00:22:13,650 --> 00:22:16,680 And in fact, we'll spend a fair amount of time thinking 443 00:22:16,680 --> 00:22:18,472 through the rules of combination. 444 00:22:22,090 --> 00:22:28,430 So that's kind of an overview of the most important thing we're 445 00:22:28,430 --> 00:22:31,310 going to talk about in this class, which is the 6.003 446 00:22:31,310 --> 00:22:33,860 abstraction, the idea of representing 447 00:22:33,860 --> 00:22:38,050 a system by the way it transforms 448 00:22:38,050 --> 00:22:42,520 an input into an output. 449 00:22:42,520 --> 00:22:44,520 So with that kind of overview, what I want to do 450 00:22:44,520 --> 00:22:49,140 next is say a little more about what are signals 451 00:22:49,140 --> 00:22:50,610 and what are systems. 452 00:22:50,610 --> 00:22:53,010 It's the signals and systems abstraction. 453 00:22:53,010 --> 00:22:56,010 We're going to need to know some more details about what is 454 00:22:56,010 --> 00:23:01,190 a signal and what is a system. 455 00:23:01,190 --> 00:23:04,390 So basically, a signal is just a mathematical function. 456 00:23:04,390 --> 00:23:08,407 In all the examples I talked about so far, 457 00:23:08,407 --> 00:23:09,990 and in many of the examples that we'll 458 00:23:09,990 --> 00:23:13,530 talk about throughout the term, the function 459 00:23:13,530 --> 00:23:16,630 is a function of time. 460 00:23:16,630 --> 00:23:19,730 And the signal can have many different dimensions. 461 00:23:19,730 --> 00:23:23,620 So for example, the mass-spring system 462 00:23:23,620 --> 00:23:25,520 evolved as a function of time. 463 00:23:25,520 --> 00:23:27,520 The tank system, the leaky tank system 464 00:23:27,520 --> 00:23:29,420 evolved as a function of time. 465 00:23:29,420 --> 00:23:31,450 The cell phone system had acoustic signals 466 00:23:31,450 --> 00:23:33,760 that were functions of time. 467 00:23:33,760 --> 00:23:36,220 Time was the same in each case. 468 00:23:36,220 --> 00:23:40,780 But the dependent variable can be a variety of things. 469 00:23:40,780 --> 00:23:43,670 Here it was position. 470 00:23:43,670 --> 00:23:50,280 So here it was flow rates, meter cubed per second of water. 471 00:23:50,280 --> 00:23:55,900 Here it was perhaps pascals, some unit of pressure 472 00:23:55,900 --> 00:23:59,260 to characterize the acoustic waveform. 473 00:23:59,260 --> 00:24:05,210 So the point is that signals are generally functions. 474 00:24:05,210 --> 00:24:09,260 We'll see that just like functions in mathematics, 475 00:24:09,260 --> 00:24:12,147 there's a lot more to functions than dependent variables 476 00:24:12,147 --> 00:24:13,230 and independent variables. 477 00:24:13,230 --> 00:24:15,104 And in fact, that's going to be a key feature 478 00:24:15,104 --> 00:24:17,630 of our analysis of signals. 479 00:24:17,630 --> 00:24:18,470 But that's to come. 480 00:24:18,470 --> 00:24:22,760 So for the time being, the simplest and complete model 481 00:24:22,760 --> 00:24:25,910 for what a signal is is it's just a function. 482 00:24:25,910 --> 00:24:28,940 The function doesn't have to be a one-dimensional function. 483 00:24:28,940 --> 00:24:32,360 In fact, a lot of the interesting applications 484 00:24:32,360 --> 00:24:33,950 of signals and systems is to look 485 00:24:33,950 --> 00:24:35,760 at multi-dimensioned functions. 486 00:24:35,760 --> 00:24:39,020 So for example, my research is in hearing. 487 00:24:39,020 --> 00:24:43,640 I'm interested to study how do the cells respond 488 00:24:43,640 --> 00:24:49,490 to sound so that we can understand how broken ears work 489 00:24:49,490 --> 00:24:51,970 differently. 490 00:24:51,970 --> 00:24:54,190 When you have a hearing deficit, like I do, 491 00:24:54,190 --> 00:24:56,680 when you have a hearing deficit, what 492 00:24:56,680 --> 00:25:00,220 is the manifestation of that at a signal-processing level? 493 00:25:00,220 --> 00:25:02,440 How do the cells respond differently 494 00:25:02,440 --> 00:25:05,710 to people who have impaired hearing from people 495 00:25:05,710 --> 00:25:08,500 who have normal hearing? 496 00:25:08,500 --> 00:25:10,330 In order to study that problem, we 497 00:25:10,330 --> 00:25:16,650 take video pictures of the cells at large magnifications 498 00:25:16,650 --> 00:25:21,740 and watch them wiggle when sounds hit them. 499 00:25:21,740 --> 00:25:24,890 So that's a picture-processing example. 500 00:25:24,890 --> 00:25:30,200 So the signals, the independent variable is not just time. 501 00:25:30,200 --> 00:25:31,190 It's a picture. 502 00:25:31,190 --> 00:25:34,174 So it might have x and y. 503 00:25:34,174 --> 00:25:36,340 In fact, the pictures we take are three dimensional. 504 00:25:36,340 --> 00:25:38,537 So it has x, y, and z. 505 00:25:38,537 --> 00:25:40,620 And in fact, they're four dimensional because they 506 00:25:40,620 --> 00:25:41,440 change with time. 507 00:25:44,920 --> 00:25:46,750 We apply this kind of a technique 508 00:25:46,750 --> 00:25:50,350 on 4D signals-- x, y, z, time. 509 00:25:50,350 --> 00:25:51,920 But it's still the property. 510 00:25:51,920 --> 00:25:54,940 It still has a property that the signals of interest 511 00:25:54,940 --> 00:25:55,870 are functions. 512 00:26:00,540 --> 00:26:05,420 We will be especially interested in this class 513 00:26:05,420 --> 00:26:10,550 in two distinct representations of signals. 514 00:26:10,550 --> 00:26:13,030 We will call them CT signals and DT signals. 515 00:26:13,030 --> 00:26:14,280 CT is continuous time. 516 00:26:14,280 --> 00:26:16,190 DT is discrete time. 517 00:26:16,190 --> 00:26:18,850 We're very interested in that because we're engineers. 518 00:26:21,380 --> 00:26:27,170 A lot of physics lives in continuous time. 519 00:26:27,170 --> 00:26:29,540 OK, the mass-spring-dashpot system, 520 00:26:29,540 --> 00:26:33,670 the signals were functions of time. 521 00:26:33,670 --> 00:26:37,375 Time was a continuously varying independent variable. 522 00:26:40,020 --> 00:26:42,710 So the leaky tank, the cell phones, those 523 00:26:42,710 --> 00:26:48,650 were all systems whose signals evolve in time. 524 00:26:48,650 --> 00:26:50,450 So they're all continuous. 525 00:26:50,450 --> 00:26:52,582 You can start with a second and break it in half 526 00:26:52,582 --> 00:26:55,040 and get a half a second and a half, and a half, and a half. 527 00:26:55,040 --> 00:26:58,850 And there's no limit to how many halves you can take. 528 00:26:58,850 --> 00:27:04,490 By contrast, a lot of the systems that we will look at 529 00:27:04,490 --> 00:27:10,530 are things that evolve in discrete time. 530 00:27:10,530 --> 00:27:12,500 What's your bank account? 531 00:27:12,500 --> 00:27:15,970 Well it only gets computed once a day. 532 00:27:15,970 --> 00:27:18,370 So it doesn't make sense to talk about your bank account 533 00:27:18,370 --> 00:27:21,140 at 9 AM and 2 PM. 534 00:27:21,140 --> 00:27:23,140 Right? 535 00:27:23,140 --> 00:27:26,470 The bank updates your account once a day. 536 00:27:26,470 --> 00:27:29,050 So it's something that happens in discrete time. 537 00:27:29,050 --> 00:27:33,240 It's something that happens in all computational systems. 538 00:27:33,240 --> 00:27:36,030 So computational systems generally 539 00:27:36,030 --> 00:27:42,950 operate on signals that are functions of discrete time. 540 00:27:42,950 --> 00:27:46,520 What was the state at time zero? 541 00:27:46,520 --> 00:27:48,410 What was the state at time one? 542 00:27:48,410 --> 00:27:51,480 What was the state at time two? 543 00:27:51,480 --> 00:27:55,890 So the system is something that eats a discrete time signal 544 00:27:55,890 --> 00:28:01,020 and generates a discrete time output signal. 545 00:28:01,020 --> 00:28:03,120 And a unique part of this class will 546 00:28:03,120 --> 00:28:05,190 be converting between the two representations. 547 00:28:05,190 --> 00:28:07,530 Because as engineers, we often want 548 00:28:07,530 --> 00:28:12,150 to build something that operates in the physical world-- 549 00:28:12,150 --> 00:28:13,650 masses and springs and reservoirs 550 00:28:13,650 --> 00:28:15,710 and that kind of stuff-- 551 00:28:15,710 --> 00:28:19,430 and do the processing computationally. 552 00:28:19,430 --> 00:28:23,080 Ever since the advent of digital electronics, 553 00:28:23,080 --> 00:28:24,980 it's just much easier. 554 00:28:24,980 --> 00:28:27,740 Digital one, it's just much easier 555 00:28:27,740 --> 00:28:33,120 to process signals in the digital domain. 556 00:28:33,120 --> 00:28:35,630 So we will often be interested in, 557 00:28:35,630 --> 00:28:39,620 how do you represent a signal who 558 00:28:39,620 --> 00:28:42,350 naturally lives in the physical world, 559 00:28:42,350 --> 00:28:44,150 and is therefore a part-- 560 00:28:44,150 --> 00:28:48,470 whose signals are continuous in nature, continuous time, 561 00:28:48,470 --> 00:28:50,720 how do you convert it into a discrete representation 562 00:28:50,720 --> 00:28:53,130 so you can crunch it on a computer? 563 00:28:53,130 --> 00:28:54,480 And in fact, how do you go back? 564 00:28:54,480 --> 00:28:57,290 So for example, if we were thinking 565 00:28:57,290 --> 00:29:00,200 about processing audio signals, we 566 00:29:00,200 --> 00:29:01,910 might want to think about how we would 567 00:29:01,910 --> 00:29:03,980 take a signal of continuous time and turn it 568 00:29:03,980 --> 00:29:06,740 into a discrete time representation. 569 00:29:06,740 --> 00:29:09,650 That's precisely what we do when we want 570 00:29:09,650 --> 00:29:12,530 to record something in MP3. 571 00:29:12,530 --> 00:29:14,060 MP3 represents a sound. 572 00:29:14,060 --> 00:29:15,500 A sound is something that I think 573 00:29:15,500 --> 00:29:19,370 of as a signal in continuous time. 574 00:29:19,370 --> 00:29:23,830 It's pascals as a function of seconds. 575 00:29:23,830 --> 00:29:27,490 But we want to represent it by a sequence of numbers 576 00:29:27,490 --> 00:29:30,340 because it's a sequence of numbers that's easy to store, 577 00:29:30,340 --> 00:29:33,520 to communicate, et cetera. 578 00:29:33,520 --> 00:29:35,890 We do the same sort of thing with images. 579 00:29:35,890 --> 00:29:39,580 The images of the ear that I talked about in my research 580 00:29:39,580 --> 00:29:44,610 are things that I think about happening at continuous space. 581 00:29:44,610 --> 00:29:46,920 Every time we look at a smaller and smaller dimension, 582 00:29:46,920 --> 00:29:49,410 we get a different answer because the image 583 00:29:49,410 --> 00:29:53,030 has no quantization in space. 584 00:29:53,030 --> 00:29:54,470 But when I crunch it, I don't want 585 00:29:54,470 --> 00:29:57,800 to work in something in continuous space. 586 00:29:57,800 --> 00:30:02,950 I want to work in a sampled version like a JPEG version. 587 00:30:02,950 --> 00:30:04,900 Similarly we want to convert back. 588 00:30:04,900 --> 00:30:08,860 I mean it would be useless for the case of the cell phone 589 00:30:08,860 --> 00:30:10,750 to convert it into a discrete representation 590 00:30:10,750 --> 00:30:13,880 and then not be able to hear it. 591 00:30:13,880 --> 00:30:16,090 So we'll think about the inverse process, 592 00:30:16,090 --> 00:30:17,470 which is reconstruction. 593 00:30:17,470 --> 00:30:19,930 If you had a discrete representation of a signal, 594 00:30:19,930 --> 00:30:23,620 how would you turn it into a continuous representation? 595 00:30:23,620 --> 00:30:27,240 And there's lots of ways people do that here's a way that we 596 00:30:27,240 --> 00:30:31,110 call zero-order hold where you convert 597 00:30:31,110 --> 00:30:35,460 every sample into a corresponding voltage 598 00:30:35,460 --> 00:30:38,910 and just hold that voltage until the next sample comes along. 599 00:30:38,910 --> 00:30:41,100 This is the representation that is most commonly 600 00:30:41,100 --> 00:30:42,225 used in things like MP3. 601 00:30:45,420 --> 00:30:49,320 By contrast, we might do something cleverer. 602 00:30:49,320 --> 00:30:53,380 We might literally extrapolate between samples. 603 00:30:53,380 --> 00:30:56,590 That is commonly done in picture processing. 604 00:30:56,590 --> 00:30:58,960 The reason we use the two different schemes is 605 00:30:58,960 --> 00:31:00,580 that psychophysically, the things that 606 00:31:00,580 --> 00:31:03,700 are important to your eyes and the things that 607 00:31:03,700 --> 00:31:06,420 are important to your ears are different. 608 00:31:06,420 --> 00:31:09,900 So you would hear the errors more 609 00:31:09,900 --> 00:31:11,760 if you did the linear interpolation. 610 00:31:11,760 --> 00:31:15,280 And you would see the errors more 611 00:31:15,280 --> 00:31:17,320 if you did the other interpolation. 612 00:31:17,320 --> 00:31:19,945 And we'll do lots of examples later on in the course 613 00:31:19,945 --> 00:31:21,070 so you see why that's true. 614 00:31:23,920 --> 00:31:27,140 OK, too much talking, not enough thinking-- 615 00:31:27,140 --> 00:31:30,920 so I've told you a lot about signals. 616 00:31:30,920 --> 00:31:35,530 Now, I'd like you to think about some signals. 617 00:31:35,530 --> 00:31:38,920 And so a common thing I'm going to do in this class 618 00:31:38,920 --> 00:31:41,230 is ask you to think about a question, 619 00:31:41,230 --> 00:31:44,530 talk to your neighbor, come to a consensus, and then vote. 620 00:31:44,530 --> 00:31:46,640 So that's what we're going to do here. 621 00:31:46,640 --> 00:31:47,890 I'm going to play some sounds. 622 00:31:47,890 --> 00:31:50,290 You're going to try to figure out what signals were 623 00:31:50,290 --> 00:31:52,120 represented by those sounds. 624 00:31:52,120 --> 00:31:56,290 But I would like you to have an opinion before we communicate 625 00:31:56,290 --> 00:31:58,750 the answer because you can prove that people 626 00:31:58,750 --> 00:32:01,550 get more out of it when there's something at stake. 627 00:32:01,550 --> 00:32:03,770 So I would like you to work with a partner. 628 00:32:03,770 --> 00:32:06,880 So in order to work with a partner, everybody stand up. 629 00:32:06,880 --> 00:32:08,557 Introduce yourself to your neighbor. 630 00:32:08,557 --> 00:32:09,640 Figure out a good partner. 631 00:32:09,640 --> 00:32:10,764 Figure out where they live. 632 00:32:10,764 --> 00:32:12,404 [CLASSROOM CHATTER] 633 00:32:56,300 --> 00:33:00,820 OK, so what I'm going to do now is play 634 00:33:00,820 --> 00:33:02,380 a computer-generated sound. 635 00:33:02,380 --> 00:33:05,260 This is a computer-generated speech sound 636 00:33:05,260 --> 00:33:08,500 made by Bob Donovan who did his PhD thesis figuring out 637 00:33:08,500 --> 00:33:11,230 how to do computer-generated speech. 638 00:33:11,230 --> 00:33:17,320 And then I'm going to play four transformations of that sound. 639 00:33:17,320 --> 00:33:20,770 And your task is to identify what's 640 00:33:20,770 --> 00:33:23,050 the mathematical representation of the transformations 641 00:33:23,050 --> 00:33:25,570 that I'm telling you. 642 00:33:25,570 --> 00:33:28,160 OK, so first, the original. 643 00:33:28,160 --> 00:33:32,283 First is the signal that Bob Donovan made. 644 00:33:32,283 --> 00:33:34,074 COMPUTER-GENERATED SPEECH: I must apologize 645 00:33:34,074 --> 00:33:35,541 for speaking [INAUDIBLE]. 646 00:33:35,541 --> 00:33:37,986 But you see, I have no brain. 647 00:33:40,754 --> 00:33:42,170 DENNIS FREEMAN: And now, I'm going 648 00:33:42,170 --> 00:33:46,400 to play a sequence of four transformations-- 649 00:33:46,400 --> 00:33:48,230 one, two, three, four. 650 00:33:48,230 --> 00:33:50,210 And the question is going to be is 651 00:33:50,210 --> 00:33:55,160 this the correct mathematical representation for this sound. 652 00:33:55,160 --> 00:33:58,910 The sound you just heard was f of t. 653 00:33:58,910 --> 00:34:03,290 Is the following sound, which by definition is f1, 654 00:34:03,290 --> 00:34:04,630 the same as f of 2t? 655 00:34:07,985 --> 00:34:09,610 COMPUTER-GENERATED SPEECH: [INAUDIBLE]. 656 00:34:15,145 --> 00:34:16,888 DENNIS FREEMAN: You're allowed to talk. 657 00:34:16,888 --> 00:34:18,864 [CLASSROOM CHATTER] 658 00:34:58,845 --> 00:35:01,530 OK, remember that answer. 659 00:35:01,530 --> 00:35:05,923 Now, is the following sound minus f of t. 660 00:35:05,923 --> 00:35:07,547 COMPUTER-GENERATED SPEECH: [INAUDIBLE]. 661 00:35:12,020 --> 00:35:14,008 [CLASSROOM CHATTER] 662 00:35:39,980 --> 00:35:42,150 DENNIS FREEMAN: OK, remember that answer. 663 00:35:42,150 --> 00:35:46,185 Is the following sound f of 2t? 664 00:35:46,185 --> 00:35:47,809 COMPUTER-GENERATED SPEECH: [INAUDIBLE]. 665 00:35:57,062 --> 00:36:02,505 [CLASSROOM CHATTER] 666 00:36:02,505 --> 00:36:05,940 DENNIS FREEMAN: And finally, is the following signal 667 00:36:05,940 --> 00:36:07,110 one third of f of t? 668 00:36:07,110 --> 00:36:08,901 COMPUTER-GENERATED SPEECH: I must apologize 669 00:36:08,901 --> 00:36:10,020 for speaking [INAUDIBLE]. 670 00:36:10,020 --> 00:36:13,650 But you see, I have no brain. 671 00:36:13,650 --> 00:36:17,290 DENNIS FREEMAN: OK, confer with your partner. 672 00:36:17,290 --> 00:36:19,090 Figure out how many are true. 673 00:36:19,090 --> 00:36:21,820 Because in 30 seconds, I'm going to ask you to raise your hand. 674 00:36:21,820 --> 00:36:23,808 [CLASSROOM CHATTER] 675 00:36:44,381 --> 00:36:44,880 Question or? 676 00:36:44,880 --> 00:36:47,122 AUDIENCE: Question, f of t [INAUDIBLE]? 677 00:36:47,122 --> 00:36:49,080 DENNIS FREEMAN: f of t was the original signal. 678 00:36:49,080 --> 00:36:53,075 AUDIENCE: I know, but like [INAUDIBLE]. 679 00:36:53,075 --> 00:36:55,380 DENNIS FREEMAN: Ah, pressure, pressure 680 00:36:55,380 --> 00:36:57,059 in the acoustic waveform. 681 00:36:57,059 --> 00:36:59,404 AUDIENCE: If you were to have minus f of t, 682 00:36:59,404 --> 00:37:01,504 your ear wouldn't [INAUDIBLE]. 683 00:37:01,504 --> 00:37:02,670 DENNIS FREEMAN: That's true. 684 00:37:05,670 --> 00:37:08,010 OK, how many of those statements are true? 685 00:37:08,010 --> 00:37:10,740 Raise your hand with some number of fingers, keeping in mind 686 00:37:10,740 --> 00:37:14,710 that you're voting having collaborated with your partner. 687 00:37:14,710 --> 00:37:16,620 So it's not your fault. You have a-- 688 00:37:16,620 --> 00:37:17,370 right, anyway. 689 00:37:17,370 --> 00:37:18,626 OK, so how many? 690 00:37:18,626 --> 00:37:19,750 Everybody raise your hands. 691 00:37:19,750 --> 00:37:20,333 I want to see. 692 00:37:24,810 --> 00:37:26,820 OK, it's about 90%-- 693 00:37:26,820 --> 00:37:29,910 no, less than that about 85% correct. 694 00:37:29,910 --> 00:37:34,773 OK, so first one, was that f of 2t? 695 00:37:34,773 --> 00:37:35,587 AUDIENCE: Yes. 696 00:37:35,587 --> 00:37:37,170 DENNIS FREEMAN: Why do you think that? 697 00:37:37,170 --> 00:37:40,320 Everybody seems to be saying yes. 698 00:37:40,320 --> 00:37:40,990 It was fast. 699 00:37:40,990 --> 00:37:41,850 It sounded faster. 700 00:37:41,850 --> 00:37:44,824 So what's faster mean from a signal point of view? 701 00:37:44,824 --> 00:37:47,167 AUDIENCE: [INAUDIBLE]. 702 00:37:47,167 --> 00:37:48,750 DENNIS FREEMAN: Change the time scale. 703 00:37:48,750 --> 00:37:50,583 That's a very good way of thinking about it. 704 00:37:50,583 --> 00:37:52,530 So if we thought about-- 705 00:37:52,530 --> 00:37:56,380 if we made a simple representation for f-- 706 00:37:56,380 --> 00:37:57,630 ignore what it really is. 707 00:37:57,630 --> 00:37:58,470 Let's say it's that. 708 00:38:00,990 --> 00:38:04,890 What would f1 look like in order to make it sound faster? 709 00:38:07,675 --> 00:38:09,944 AUDIENCE: [INAUDIBLE]. 710 00:38:09,944 --> 00:38:11,860 DENNIS FREEMAN: OK, we have people doing this. 711 00:38:11,860 --> 00:38:13,110 And we have people doing this. 712 00:38:13,110 --> 00:38:15,530 And I'm not quite sure what that means. 713 00:38:15,530 --> 00:38:18,580 So could somebody be a little more descriptive? 714 00:38:18,580 --> 00:38:20,620 Hand gestures are fine. 715 00:38:20,620 --> 00:38:21,360 Yes. 716 00:38:21,360 --> 00:38:22,502 AUDIENCE: Squished. 717 00:38:22,502 --> 00:38:24,620 DENNIS FREEMAN: Squished, exactly right. 718 00:38:24,620 --> 00:38:29,920 So what we want to have happen, in order to sound faster, 719 00:38:29,920 --> 00:38:32,320 we want to have it come out faster. 720 00:38:32,320 --> 00:38:35,420 So we would like that to happen to it. 721 00:38:35,420 --> 00:38:39,100 So it's all over in one second if the original 722 00:38:39,100 --> 00:38:40,550 was all over in two seconds. 723 00:38:40,550 --> 00:38:41,050 Right? 724 00:38:41,050 --> 00:38:42,910 That's what faster means. 725 00:38:42,910 --> 00:38:44,830 So we would like that to be true. 726 00:38:44,830 --> 00:38:47,682 Is that what's going on in that function? 727 00:38:47,682 --> 00:38:49,490 Yeah. 728 00:38:49,490 --> 00:38:51,490 OK, so that's right. 729 00:38:51,490 --> 00:38:54,380 Secondly-- yes. 730 00:38:54,380 --> 00:38:56,957 AUDIENCE: Is that the reason also why it was higher pitched? 731 00:38:56,957 --> 00:38:59,040 DENNIS FREEMAN: That also makes it higher pitched. 732 00:38:59,040 --> 00:39:02,210 In fact, we will talk a lot about this later. 733 00:39:02,210 --> 00:39:06,260 But pitch has to do with a related kind of waveform. 734 00:39:06,260 --> 00:39:11,870 If I thought about a waveform that was a single tone, 735 00:39:11,870 --> 00:39:15,070 that might look like this. 736 00:39:15,070 --> 00:39:20,800 So if you were to play an oboe and play just a C constantly, 737 00:39:20,800 --> 00:39:24,130 you'd get some periodic waveform. 738 00:39:24,130 --> 00:39:27,610 If I did the same transformation on that periodic waveform, 739 00:39:27,610 --> 00:39:29,800 what would happen to that waveform? 740 00:39:29,800 --> 00:39:35,940 What would happen to this if I did transformation one? 741 00:39:35,940 --> 00:39:40,030 Squish, what's squish mean? 742 00:39:40,030 --> 00:39:43,171 More higher frequency, more cycles per second. 743 00:39:43,171 --> 00:39:43,670 Yeah. 744 00:39:43,670 --> 00:39:46,205 AUDIENCE: So how can you make the same waveform go twice as 745 00:39:46,205 --> 00:39:48,445 fast, but have the same pitch [INAUDIBLE]? 746 00:39:48,445 --> 00:39:49,910 DENNIS FREEMAN: That's very hard. 747 00:39:49,910 --> 00:39:53,390 In fact, I worked on that as a research project. 748 00:39:53,390 --> 00:39:55,540 So one of the techniques that we used 749 00:39:55,540 --> 00:39:57,505 to try to fix people's ears-- 750 00:40:00,310 --> 00:40:01,990 it turns out that people who have 751 00:40:01,990 --> 00:40:06,640 hearing disorders like mine can more easily understand 752 00:40:06,640 --> 00:40:08,920 male speech than female speech just because 753 00:40:08,920 --> 00:40:10,270 of the shift in frequencies. 754 00:40:10,270 --> 00:40:14,050 Male speech is primarily about an octave lower 755 00:40:14,050 --> 00:40:16,330 on average than female speech. 756 00:40:16,330 --> 00:40:19,090 So we tried to convert every female 757 00:40:19,090 --> 00:40:21,970 into a male speaker and every male 758 00:40:21,970 --> 00:40:26,470 into the Jolly Green Giant. 759 00:40:26,470 --> 00:40:27,760 And it worked. 760 00:40:27,760 --> 00:40:30,340 Males sounded like the Jolly Green Giant. 761 00:40:30,340 --> 00:40:32,440 And females sounded like males. 762 00:40:32,440 --> 00:40:34,420 And it didn't help hearing at all. 763 00:40:34,420 --> 00:40:38,195 And that's primarily because it's a hard problem. 764 00:40:38,195 --> 00:40:40,820 But we will say more about that as we go on through the course. 765 00:40:40,820 --> 00:40:44,560 That is a hard problem because you end up 766 00:40:44,560 --> 00:40:47,860 having to do other than just squish. 767 00:40:47,860 --> 00:40:52,930 OK, second one, is this is the right transformation? 768 00:40:52,930 --> 00:40:55,628 No, why not? 769 00:40:55,628 --> 00:40:58,025 AUDIENCE: [INAUDIBLE] so most of the speech 770 00:40:58,025 --> 00:40:59,524 is measured in the frequency domain. 771 00:40:59,524 --> 00:41:01,954 But if you just flip over the amplitude, 772 00:41:01,954 --> 00:41:03,495 it shouldn't really make a difference 773 00:41:03,495 --> 00:41:06,060 DENNIS FREEMAN: If you flipped over the amplitude-- 774 00:41:06,060 --> 00:41:08,070 so flip over the amplitude, you're 775 00:41:08,070 --> 00:41:09,900 saying that didn't do it. 776 00:41:09,900 --> 00:41:11,700 What was the transformation? 777 00:41:11,700 --> 00:41:14,370 If it wasn't flip over the amplitude, what is it? 778 00:41:14,370 --> 00:41:15,865 Yeah. 779 00:41:15,865 --> 00:41:17,325 AUDIENCE: f of minus t. 780 00:41:17,325 --> 00:41:18,580 DENNIS FREEMAN: f of minus t-- 781 00:41:18,580 --> 00:41:21,280 it was flipped this way. 782 00:41:21,280 --> 00:41:25,420 Right, flipping this way is largely inaudible. 783 00:41:25,420 --> 00:41:28,030 If you had a pressure waveform versus a rarefaction waveform, 784 00:41:28,030 --> 00:41:29,982 positive pressures versus negative pressures, 785 00:41:29,982 --> 00:41:30,940 it's very hard to tell. 786 00:41:30,940 --> 00:41:34,879 Not impossible, if you're one of those audio types like me, 787 00:41:34,879 --> 00:41:36,420 you can probably hear the difference. 788 00:41:36,420 --> 00:41:38,580 But it's very hard. 789 00:41:38,580 --> 00:41:41,510 Hearing this, not hard. 790 00:41:41,510 --> 00:41:43,020 OK, how about this one? 791 00:41:45,930 --> 00:41:49,740 Obviously wrong because this one was right. 792 00:41:49,740 --> 00:41:51,960 OK, how about this one? 793 00:41:51,960 --> 00:41:52,950 Yeah. 794 00:41:52,950 --> 00:41:57,760 So the idea was that there were two that were correct. 795 00:41:57,760 --> 00:42:01,124 So the 85% of you or so who got two, presumably 796 00:42:01,124 --> 00:42:02,040 you got the right two. 797 00:42:02,040 --> 00:42:02,550 Right? 798 00:42:02,550 --> 00:42:05,110 But I won't ask. 799 00:42:05,110 --> 00:42:09,600 OK, here's an image processing question. 800 00:42:09,600 --> 00:42:12,740 Think about this picture of Stata. 801 00:42:12,740 --> 00:42:15,050 I know it's hard to think about pictures of Stata. 802 00:42:15,050 --> 00:42:17,034 That's OK. 803 00:42:17,034 --> 00:42:18,450 Think about this picture is Stata. 804 00:42:18,450 --> 00:42:21,470 I have indexed x and y. 805 00:42:21,470 --> 00:42:24,260 And I've written that as f, a signal that 806 00:42:24,260 --> 00:42:29,480 depends on two independent variables, x and y. 807 00:42:29,480 --> 00:42:32,170 And I've got three transformations-- 808 00:42:32,170 --> 00:42:34,660 the f1 transformation, the f2, and the f3. 809 00:42:37,180 --> 00:42:40,510 The question is, is this the right mathematical 810 00:42:40,510 --> 00:42:47,520 representation for transforming that picture into this one? 811 00:42:47,520 --> 00:42:49,075 Is this the one for that one? 812 00:42:49,075 --> 00:42:51,810 And is this a representation for this one? 813 00:42:51,810 --> 00:42:53,790 Is this the representation for that one? 814 00:42:53,790 --> 00:42:56,070 Take 30 seconds, talk to your partner, 815 00:42:56,070 --> 00:42:59,348 and figure out how many of those transformations are correct. 816 00:42:59,348 --> 00:43:01,344 [CLASSROOM CHATTER] 817 00:44:17,765 --> 00:44:19,890 OK, so how many of the transformations are correct? 818 00:44:19,890 --> 00:44:22,782 Raise your hand. 819 00:44:22,782 --> 00:44:23,865 Everybody raise your hand. 820 00:44:28,209 --> 00:44:30,310 That's a zero or a three? 821 00:44:30,310 --> 00:44:32,770 OK, just checking. 822 00:44:32,770 --> 00:44:35,980 OK, about 85% correct again. 823 00:44:35,980 --> 00:44:39,400 OK, probably a different 85%. 824 00:44:39,400 --> 00:44:41,300 So how do you think about this problem? 825 00:44:41,300 --> 00:44:45,640 So there's a variety of ways you could think about it. 826 00:44:45,640 --> 00:44:47,560 Let me show you a way that focuses 827 00:44:47,560 --> 00:44:49,930 on an idea called mapping because I 828 00:44:49,930 --> 00:44:51,950 think it's a very powerful way. 829 00:44:51,950 --> 00:44:55,900 So if you think about a map, how does the t 830 00:44:55,900 --> 00:44:57,910 variable in the first problem map 831 00:44:57,910 --> 00:45:00,880 to the t variable in the second problem? 832 00:45:00,880 --> 00:45:02,320 So a way you can think about this 833 00:45:02,320 --> 00:45:08,500 is if this is true for all x and y, it's true 834 00:45:08,500 --> 00:45:10,690 for a particular x. 835 00:45:10,690 --> 00:45:12,800 So let's ask is it true for x equals 0. 836 00:45:15,350 --> 00:45:19,710 So how does the point x equals 0 map from one image to another? 837 00:45:19,710 --> 00:45:24,240 So if you substituted x equals 0, then f1 of 0-- 838 00:45:24,240 --> 00:45:28,360 so if you look at this transformation, 839 00:45:28,360 --> 00:45:35,860 the claim would be that f1 of 0 is the same as f of 0. 840 00:45:35,860 --> 00:45:36,490 Is that true? 841 00:45:42,440 --> 00:45:47,090 So f1 of 0 bisects Stata. 842 00:45:49,840 --> 00:45:53,990 f of 0 bisects Stata. 843 00:45:53,990 --> 00:45:56,120 So yeah, that's the right thing. 844 00:45:56,120 --> 00:45:58,900 Everybody see that? 845 00:45:58,900 --> 00:46:02,650 Is the statement true for x equals 250? 846 00:46:02,650 --> 00:46:05,971 Well, f1 of 250-- 847 00:46:05,971 --> 00:46:11,980 so f1 of 250 is through the right-hand side 848 00:46:11,980 --> 00:46:18,860 of Stata sort of where the steel and the bricks come together. 849 00:46:18,860 --> 00:46:22,940 Is that the same as substituting in here, if x is 850 00:46:22,940 --> 00:46:26,000 250, then f of 500-- 851 00:46:26,000 --> 00:46:29,780 f of 500 is off the screen. 852 00:46:29,780 --> 00:46:34,370 So is that transformation hold at the point x equals 250? 853 00:46:34,370 --> 00:46:35,240 No. 854 00:46:35,240 --> 00:46:36,990 OK, it can't be right. 855 00:46:36,990 --> 00:46:37,910 OK? 856 00:46:37,910 --> 00:46:40,550 One bad sample proves that it can't 857 00:46:40,550 --> 00:46:43,430 be the general transformation. 858 00:46:43,430 --> 00:46:46,250 OK, everybody see that? 859 00:46:46,250 --> 00:46:48,230 Similarly, for the second transformation, 860 00:46:48,230 --> 00:46:50,480 let's just try some points. 861 00:46:50,480 --> 00:46:57,050 If we try the point x equals 0, this says that f2 at location 0 862 00:46:57,050 --> 00:47:00,804 should be f at location minus 250. 863 00:47:00,804 --> 00:47:01,470 Is that correct? 864 00:47:06,640 --> 00:47:13,560 So f of 0, well, that's the left-hand side of Stata. 865 00:47:13,560 --> 00:47:17,310 And so f2 of 0, that's the left-hand side of Stata. f 866 00:47:17,310 --> 00:47:23,470 of 2 times 0 minus 250 is f of 250 minus 250, 867 00:47:23,470 --> 00:47:26,230 that's also the left of Stata. 868 00:47:26,230 --> 00:47:28,220 OK, so by using that reasoning, you 869 00:47:28,220 --> 00:47:33,860 can go through and see if certain points map 870 00:47:33,860 --> 00:47:36,420 to the corresponding place where they were supposed to go. 871 00:47:36,420 --> 00:47:39,060 That's the idea of mapping. 872 00:47:39,060 --> 00:47:45,650 And in this particular case, if we traced two points, 873 00:47:45,650 --> 00:47:50,150 and the map 2x minus 250 is a linear function 874 00:47:50,150 --> 00:47:54,520 of x, so two points determine a straight line. 875 00:47:54,520 --> 00:47:58,720 So after you've determined two points, you're done. 876 00:47:58,720 --> 00:48:00,840 Seem right? 877 00:48:00,840 --> 00:48:04,170 So similarly, you can figure out that the particular points 878 00:48:04,170 --> 00:48:04,980 didn't work here. 879 00:48:04,980 --> 00:48:07,460 And so the answer is that only one of those transformations 880 00:48:07,460 --> 00:48:07,960 was right. 881 00:48:10,860 --> 00:48:15,790 OK, so I spent a fair amount of time on signals. 882 00:48:15,790 --> 00:48:18,040 In the last minute, oh, minute, no, 883 00:48:18,040 --> 00:48:20,524 I'm not going to do that am I. OK, 884 00:48:20,524 --> 00:48:22,690 I spent a fair amount of time talking about signals. 885 00:48:22,690 --> 00:48:28,210 Next time, we'll talk about similar ways of thinking 886 00:48:28,210 --> 00:48:31,540 about simple systems. 887 00:48:31,540 --> 00:48:34,560 So have a good weekend.